"There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds." G.H. Hardy, A Mathematician's Apology (1941)

Fundamental principle behind many dynamical systems verified

Diffusion is a physical process that is an essential component of innumerable phenomena in nature, including when your lungs transfer oxygen into and carbon dioxide out of your blood. Now a fundamental principle behind diffusion has at last been experimentally verified.

An ergodic dynamical system is one for which the average behaviour of a single component over time is the same as the average for all components at a single time. This allows researchers to assume for some systems that measurement of a given variable on one particle – such as the distance covered by a particle in a given time interval – should yield the same average value as a single measurement of the same variable on a collection of particles, so that each particle behaves as a small version of the system as a whole.

It is generally accepted that the ergodic hypothesis is applicable to the dynamics of diffusive processes. Diffusion describes the spread of particles through random motion under the influence of thermal energy. In virtually all chemical reactions, diffusion is responsible for bringing reactants sufficiently close together to enable them to react.

However, as Professor Jörg Kärger at Leipzig University points out, “although diffusive processes have been investigated for the past 150 years, the principle of ergodicity has not yet been experimentally verified”. This is because so far it has only been possible to quantify diffusive processes by measuring many particles simultaneously.

A collaborative effort by Professor Kärger’s group at Leipzig University and Professor Christoph Bräuchle’s team at LMU Munich looked to measure the diffusive behaviour of a system of particles and the trajectories of single molecules in the same system. The LMU researchers were able to track individual molecules, while the Leipzig group could study the collective behaviour of the whole system.

The problem for the teams to overcome was that successful application of the two methods requires apparently conflicting conditions. The Leipzig group’s method needed high concentrations of molecules with large diffusion coefficients, while the LMU team’s method works best with extremely dilute solutions of species with small diffusion coefficients.

Using particular organic dyes with high fluorescence yields in combination with porous silicate glasses containing networks of nanometer-sized channels in which the dye molecules can diffuse, the researchers were able to create a system on which both techniques could be performed.

Comparing data, the two teams found that the diffusion coefficients obtained by the two techniques agreed with each other, providing the first experimental confirmation of the ergodic hypothesis in this context.

This work is important because sometimes a mathematical theory can make accurate predictions about the real world up to a point but not actually reflect the real world context. A nice example is from cosmology in the late 16th Century, when mathematical models of the universe with the Earth or the Sun at the centre both produced viable predictions of astronomical events like eclipses. Further experimental data was needed to demonstrate that one reflected physical reality more accurately than the other. In this way, experimental checking of fundamental principles, such as the current work at the Large Hadron Collider, is important whether or not it backs current theory.

The next step for this research will be to look at systems that do not conform to the ergodic principle and determine the reasons for this. “The diffusion of nanoparticles in cells looks like an interesting example,” says Bräuchle, “and for us the important thing is to find out why the ergodic theorem doesn’t hold in this case.”